Quantum Mechanics I: Harmonic Oscillator and Operator Formalism, Assignments of Quantum Mechanics

Download Quantum Mechanics I: Harmonic Oscillator and Operator Formalism and more Assignments Quantum Mechanics in PDF only on Docsity! PHY-851 QUANTUM MECHANICS I Homework 10, 50 points November 14 - 28, 2001 Harmonic oscillator. Operator formalism. Reading: Merzbacher, Chapter 5. 1. /7/ Merzbacher, Problem 3, p. 90. 2. /13/ Merzbacher, Problem 6, p. 91. 3. /6/ A particle of mass m and electric charge e is placed in the one- dimensional harmonic oscillator potential of frequency ω and the uniform electric field E along the same axis. a. Find the wave functions and the energy spectrum of the particle. b. With a particle in the ground state of the problem, at time t = 0 the electric field is suddenly switched off. Find the probability of finding the particle at t > 0 in the nth stationary state of the oscillator. c. The ground state in the presence of the electric field acquires the nonzero expectation value of the electric dipole moment 〈d̂〉 proportional to the applied field E . Find the coefficient of proportionality (static po- larizability). 4. /15/ Any linear operator F̂ in the coordinate representation can be defined as an integral operator acting on an arbitrary function ψ(x) according to F̂ψ(x) = ∫ dx′ F (x, x′)ψ(x′), (1) where the function F (x, x′) is called the kernel of the operator. a. Construct kernels F (x, x′) corresponding to the operators x̂, p̂, inver- sion P̂, displacement D̂(a) and scale transformation M̂(α); the last three operators are defined in Problem 3, Homework 4. b. Find the most general form of the kernel F (x, x′) for an operator F̂ which commutes with the coordinate operator x̂. c. Find the most general form of the kernel F (x, x′) for an operator F̂ which commutes with the momentum operator p̂. d. Find the most general form of the kernel F (x, x′) for an operator F̂ which commutes with x̂ and p̂. e. Consider an operator F̂ with the factorized kernel, F (x, x′) = f(x)g(x′). At what condition the operator F̂ is Hermitian? For a Hermitian operator of this type find its eigenfunctions and eigenvalues. Find the degeneracies of the eigenvalues (their multiplicities in the spectrum). 1